Complex varieties (smooth or singular) are very rich geometric objects that can be studied from both the analytical and algebraic point of view. We will first introduce complex varieties of arbitrary dimension as well as the basic tools needed for their study. In the second part of the course we study in more detail the case of curves. This course provides the prerequisites for the courses of F. Labourie and Dimca-Parusinski. Topics of the first part are holomorphic functions in several variables, analytic sets in C^n, complex manifolds, vector bundles, differential forms of type (p, q), Dolbeault and de Rham cohomology, Riemann-Roch theorem on a compact Riemann surface. Topics of the second part are definition of Riemann surfaces, basic examples, associated algebraic curves and Riemann surfaces, Riemann surfaces associated with an analytic function, Riemann surfaces obtained as quotients by a group of discrete automorphismes, Riemannian metrics on surfaces, conformal transformations, hyperbolic geometry.